arXiv subjects

Joseph Katz

Publications and source records attributed to Joseph Katz.

At least 37 records · Page 2Linked to original sources

arXiv2008

Cosmological perturbation theory, instantaneous gauges and local inertial frames

Linear perturbations of Friedmann-Robertson-Walker universes with any curvature and cosmological constant are studied in a general gauge without decomposition into harmonics. Desirable gauges are selected as those which embody best Mach's principle: in these gauges local inertial frames can be determined instantaneously via the perturbed Einstein field equations from the distributions of energy and momentum in the universe. The inertial frames are identified by their `accelerations and rotations' with respect to the cosmological frames associated with the `Machian gauges'. In closed spherical universes, integral gauge conditions are imposed to eliminate motions generated by the conformal Killing vectors. The meaning of Traschen's integral constraint vectors is thus elucidated. For all three types of FRW universes the Machian gauges admit much less residual freedom than the synchronous or generalized harmonic gauge. Mach's principle is best exhibited in the Machian gauges in closed spherical universes. Independent of any Machian motivation the general perturbation equations and discussion of gauges are useful for cosmological perturbation theory.

gr-qc↗

arXiv2007

Energy and angular momentum densities of stationary gravity fields

We give physical explanations of explicit invariant expressions for the energy and angular momentum densities of gravitational fields in stationary space-times. These expressions involve non-locally defined conformal factors. In certain coordinates these become locally defined in terms of the metric. These results are derived via expressions for total gravitational potential energy from the difference between the total energy and the mechanical energy. The latter involves kinetic energy seen in the frame of static observers. When in the axially symmetric case we consider zero angular momentum observers (who move orthogonally to surfaces of constant time), we find that the angular momentum they attribute to the gravitational field is solely due to their motion.

gr-qc↗

arXiv2006

Gravitational energy in stationary spacetimes

Static observers remain on Killing-vector world lines and measure the rest-mass+kinetic energies of particles moving past them, and the flux of that mechanical energy through space and time. The total mechanical energy is the total flux through a spacelike cut at one time. The difference between the total mass-energy and the total mechanical energy is the total gravitational energy, which we prove to be negative for certain classes of systems. For spherical systems, Misner, Thorne and Wheeler define the total gravitational energy this way. To obtain the gravitational energy density analogous to that of electromagnetism we first use Einstein's equations with integrations by parts to remove second order derivatives. Next we apply a conformal transformation to reexpress the scalar 3-curvature of the 3-space. The resulting density is non-local. We repeat the argument for mechanical energies as measured by stationary observers moving orthogonally to constant time slices like the "zero angular momentum" observers of Bardeen who exist even within ergospheres.

gr-qc↗

arXiv2006

Comments on conformal masses, asymptotics backgrounds and conservation laws

The "conformal mass prescriptions" were used recently to calculate the mass of spacetimes in higher dimensional and higher curvature theories of gravity. These definitions are closely related to Komar integrals for spacetimes that are conformally flat at great distances from the sources. We derive these relations without using the conformal infinity formalism.

gr-qc↗

arXiv2005

Gravitational energy

Observers at rest in a stationary spacetime flat at infinity can measure small amounts of rest-mass+internal energies+kinetic energies+pressure energy in a small volume of fluid attached to a local inertial frame. The sum of these small amounts is the total "matter energy" for those observers. The total mass-energy minus the matter energy is the binding gravitational energy. Misner, Thorne and Wheeler evaluated the gravitational energy of a spherically symmetric static spacetime. Here we show how to calculate gravitational energy in any static and stationary spacetime for isolated sources with a set of observers at rest. The result of MTW is recovered and we find that electromagnetic and gravitational 3-covariant energy densities in conformastatic spacetimes are of opposite signs. Various examples suggest that gravitational energy is negative in spacetimes with special symmetries or when the energy-momentum tensor satisfies usual energy conditions.

gr-qc↗

arXiv2005

Relativistic conservation laws and integral constraints for large cosmological perturbations

For every mapping of a perturbed spacetime onto a background and with any vector field $ξ$ we construct a conserved covariant vector density $I(ξ)$, which is the divergence of a covariant antisymmetric tensor density, a "superpotential". $ I(ξ)$ is linear in the energy-momentum tensor perturbations of matter, which may be large; $I(ξ)$ does not contain the second order derivatives of the perturbed metric. The superpotential is identically zero when perturbations are absent. By integrating conserved vectors over a part $\Si$ of a hypersurface $S$ of the background, which spans a two-surface $\di\Si$, we obtain integral relations between, on the one hand, initial data of the perturbed metric components and the energy-momentum perturbations on $\Si$ and, on the other hand, the boundary values on $\di\Si$. We show that there are as many such integral relations as there are different mappings, $ξ$'s, $\Si$'s and $\di\Si$'s. For given boundary values on $\di\Si$, the integral relations may be interpreted as integral constraints (e.g., those of Traschen) on local initial data including the energy-momentum perturbations. Conservation laws expressed in terms of Killing fields $\Barξ$ of the background become "physical" conservation laws. In cosmology, to each mapping of the time axis of a Robertson-Walker space on a de Sitter space with the same spatial topology there correspond ten conservation laws. The conformal mapping leads to a straightforward generalization of conservation laws in flat spacetimes. Other mappings are also considered. ...

gr-qc↗

arXiv2005

On the uniqueness of harmonic coordinates

Harmonic coordinate conditions in stationary asymptotically flat spacetimes with matter sources have more than one solution. The solutions depend on the degree of smoothness of the metric and its first derivatives, which we wish to impose across the material boundary, and on the conditions at infinity and at a suitable point inside the matter. This is illustrated in detail by simple fully solvable examples of static spherically symmetric spacetimes in global harmonic coordinates. Examples of stationary electrovacuum spacetimes described simply in harmonic coordinates are also given. They can represent the exterior fields of material discs. The use of an appropriate background metric considerably simplifies the calculations.

gr-qc↗

arXiv2004

On the mass of a Kerr-anti-de Sitter spacetime in D dimensions

We show how to compute the mass of a Kerr-anti-de Sitter spacetime with respect to the anti-de Sitter background in any dimension, using a superpotential which has been derived from standard Noether identities. The calculation takes no account of the source of the curvature and confirms results obtained for black holes via the first law of thermodynamics.

gr-qc↗

arXiv2004

Conserved Charges in Einstein Gauss-Bonnet theory

Using Noether's identities, we define a superpotential with respect to a background for the Einstein Gauss-Bonnet theory of gravity. As an example, we show that its associated conserved charge yields the mass-energy of a D-dimensional Gauss-Bonnet black hole in an anti-de Sitter spacetime.

gr-qc↗

arXiv2003

Do Rotations Beyond the Cosmological Horizon Affect the Local Inertial Frame?

If perturbations beyond the horizon have the velocities prescribed everywhere then the dragging of inertial frames near the origin is suppressed by an exponential factor. However if perturbations are prescribed in terms of their angular momenta there is no such suppression. We resolve this paradox and in doing so give new explicit results on the dragging of inertial frames in closed, flat and open universe with and without a cosmological constant.

gr-qc↗

arXiv2003

Toroidal Perturbations of Friedmann-Robertson-Walker Universes

Explicit expressions are found for the axisymmetric metric perturbations of the closed, flat and open FRW universes caused by toroidal motions of the cosmic fluid. The perturbations are decomposed in vector spherical harmonics on 2-spheres, but the radial dependence is left general. Solutions for general odd-parity $l$-pole perturbations are given for either angular velocities or angular momenta prescribed. In particular, in case of closed universes the solutions require a special treatment of the Legendre equation.

gr-qc↗

arXiv2002

Thermodynamics and Self-Gravitating Systems

This work assembles some basic theoretical elements on thermal equilibrium, stability conditions, and fluctuation theory in self-gravitating systems illustrated with a few examples. Thermodynamics deals with states that have settled down after sufficient time has gone by. Time dependent phenomena are beyond the scope of this paper. While thermodynamics is firmly rooted in statistical physics, equilibrium configurations, stability criteria and the destabilizing effect of fluctuations are all expressed in terms of thermodynamic functions. The work is not a review paper but a pedagogical introduction which may interest theoreticians in astronomy and astrophysicists. It contains sufficient mathematical details for the reader to redo all calculations. References are only to seminal works or readable reviews. (Except in the ultimate sub-section.) Delicate mathematical problems are mentioned but are not discussed in detail.

astro-ph↗

arXiv2001

Gravity on branes

We consider the four dimensional discontinuity generated by two identical pieces of a five dimensional space pasted along their edge (that is a "brane" in a "$Z_2$-symmetric" "bulk"). Using a four plus one decomposition of the Riemann tensor we write the equations for gravity on the brane and recover in a simple manner a number of known "brane world" scenarios. We study under which conditions these equations reduce, exactly or approximately, to the four dimensional Einstein equations. We conclude that if the bulk is imposed to be only an Einstein space near the brane, Einstein's equations can be recovered approximately on the brane, but if it is imposed to be strictly anti-de Sitter space then the Einstein equations cannot hold, even approximately, on a quasi-Minkowskian brane, unless matter obeys a very contrived equation of state.

gr-qc↗

arXiv2001

Perturbations of brane worlds

We consider cosmological models where the universe, governed by Einstein's equations, is a piece of a five dimensional double-sided anti-de Sitter spacetime (that is, a "$Z_2$-symmetric bulk") with matter confined to its four dimensional Robertson-Walker boundary or "brane". We study the perturbations of such models. We use conformally minkowskian coordinates to disentangle the contributions of the bulk gravitons and of the motion of the brane. We find the restrictions put on the bulk gravitons when matter on the brane is taken to be a scalar field and we solve in that case the brane perturbation equations.

hep-th↗

arXiv2000

Fluctuations in Isothermal Spheres

Isolated isothermal spheres of N gravitationally interacting points with equal mass are believed to be stable when density contrasts do not exceed 709. That stability limit does, however, not take into consideration fluctuations of temperature near the onset of instability. These are important when N is finite. Here we correlate {\it global mean quadratic temperature fluctuations} with onset of instability. We show that such fluctuations trigger instability when the density contrast reaches a value near $709\cdot\exp(-3.3N^{-1/3})$. These lower values of limiting density contrasts are significantly smaller than 709 when N is not very big and this suggests (i) that numerical calculations with small N may not reflect correctly the onset of core collapse in clusters with big N and (ii) that a greater number of globular clusters than is normally believed may already be in an advanced stage of core collapse because most of observed globular clusters whose parameters fit quasi-isothermal configurations are close to marginal stability.

astro-ph↗

arXiv1996

On Global Conservation Laws at Null Infinity

The ``standard'' expressions for total energy, linear momentum and also angular momentum of asymptotically flat Bondi metrics at null infinity are also obtained from differential conservation laws on asymptotically flat backgrounds, derived from a quadratic Lagrangian density by methods currently used in classical field theory. It is thus a matter of taste and commodity to use or not to use a reference spacetime in defining these globally conserved quantities. Backgrounds lead to Nœther conserved currents; the use of backgrounds is in line with classical views on conservation laws. Moreover, the conserved quantities are in principle explicitly related to the sources of gravity through Einstein's equations, while standard definitions are not. The relations depend, however, on a rule for mapping spacetimes on backgrounds.

gr-qc↗

arXiv1996

Integral Constraints On cosmological Perturbations and their Energy

We show the relation between Traschen's integral equations and the energy, and ``position of the centre of mass'', of the matter perturbations in a Robertson-Walker spacetime. When the perturbations are ``localised'' we get a set of integral constraints that includes hers. We illustrate them on a simple example.

gr-qc↗

arXiv1995

Energy Principles for Self-Gravitating Barotropic Flows: I. General Theory

The following principle of minimum energy may be a powerful substitute to the dynamical perturbation method, when the latter is hard to apply. Fluid elements of self-gravitating barotropic flows, whose vortex lines extend to the boundary of the fluid, are labelled in such a way that any change of trial configurations automatically preserves mass and circulation. The velocity field is given by a mass conserving Clebsch representation. With three independent Lagrangian functions, the total energy is stationary for all small variations about a flow with fixed linear and angular momenta provided Euler's equations for steady motion are satisfied. Thus, steady flows are stable if their energy is minimum. Since energy is here minimized subject to having local and global contants of the motion fixed, stability limits obtained that way are expected to be close to limits given by dynamical perturbation methods. Moreover, the stability limits are with respect to arbitrary, not necessary small, perturbations. A weaker form of the energy principle is also given which may be easier to apply. The Lagrangian functional, with the same three Lagrange variables is stationary for the fully time dependent Euler equations. It follows that the principle of minimum energy gives stability conditions that are both necessary and sufficient if terms linear in time derivatives (gyroscopic terms) are absent from the Lagrangian. The gyroscopic term for small deviations around steady flows is given explicitly. Key words: Energy variational principle; Self-gravitating systems; Stability of fluids.

astro-ph↗